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If \sin ^2 A=x and \prod_{r=1}^4 \sin (r A)=a x^2+b x^3+c x^4+d x^5, then the value of 10a-7b+15c-5d must be 

Option: 1

3020

 


Option: 2

3428


Option: 3

3120

 


Option: 4

3448


Answers (1)

best_answer

\because \prod_{r=1}^4 \sin (r A)=\sin A \sin 2 A \sin 3 A \sin 4 A

                             \begin{aligned} & =\sin A \cdot 2 \sin A \cos A \cdot(3 \sin A \left.-4 \sin ^3 A \cdot 2 \sin 2 A \cos 2 A\right) \end{aligned}

                              \begin{aligned} & =2 \sin ^2 A \cos A \cdot \sin A\left(3-4 \sin ^2 A\right) \\ &\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 4 \sin A \cos A \cdot\left(1-2 \sin ^2 A\right) \\ & \end{aligned}

                             \begin{aligned} & =8 x^2(1-x)(3-4 x)(1-2 x) \\ \\& =24 x^2-104 x^3+144 x^4-64 x^5 \end{aligned}

On comparing, we get

a=24, b=-104, c=144, d=-64

10 a-7 b+15 c-5 d

                     \begin{aligned} & =10 \times 24-7 \times-104+15 \times 144-5 \times-64 \\ \\& =240+728+2160+320=3448 \end{aligned}

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Sayak

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